nnCostFunction.m

function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the
 
 neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into
 
 the vector
%   nn_params and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2
 
 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1);

% You need to return the following variables correctly 
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the 
%               first time.
%
% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%

% Part 1:計算J
% X=[ones(m,1) X];
% for i=1:m,
%     z2=Theta1*X(i,:)';
%     a2=sigmoid(z2);
%     a2=[1;a2];
%     z3=Theta2*a2;
%     a3=sigmoid(z3);
%     J=J+sum(log(1-a3))+log(a3(y(i,:)))-log(1-a3(y(i,:)));%由於輸出為10維向量，而y的值是1-10的數字，所以可以用y的值指示a3那些元素加，哪些不加
% end
% J=-(J/m);
% 
% 
% % Part 2: compute the gradients
% temp=0;
% for i=1:hidden_layer_size,
%     for j=2:(input_layer_size+1),
%         temp=temp+Theta1(i,j)^2;%對Theta1除了第一列（與偏置神經元對應的那列）元素的平方求和
%     end
% end
% for i=1:num_labels,
%     for j=2:(hidden_layer_size+1),
%         temp=temp+Theta2(i,j)^2;%對Theta2除了第一列（與偏置神經元對應的那列）元素的平方求和
%     end
% end
% 
% J=J+lambda/(2*m)*temp;
% 
% 
% %利用反向傳播法求取偏導數值，實際上這個迴圈可以和計算J值得迴圈合為一個，為了程式碼清晰，所以分開寫了??
% delta3=zeros(num_labels,1);%反向傳播，輸出層的誤差??
% delta2=zeros(size(Theta1));%反向傳播，隱藏層的誤差；輸入層不計算誤差??
% for i=1:m,%m為訓練樣本數，利用for遍歷?
%     a1=X(i,:)';
%     z2=Theta1*a1;
%     a2=sigmoid(z2);
%     a2=[1;a2];
%     z3=Theta2*a2;
%     a3=sigmoid(z3);
%     delta3=a3;
%     delta3(y(i,:))=delta3(y(i,:))-1;
%     delta2=Theta2'*delta3.*[1;sigmoidGradient(z2)];
%     delta2=delta2(2:end);
%     Theta2_grad=Theta2_grad+delta3*a2';
%     Theta1_grad=Theta1_grad+delta2*a1';
% end
% 
% Theta2_grad=Theta2_grad/m+lambda/m*Theta2;%正則化，修正梯度值
% Theta2_grad(:,1)=Theta2_grad(:,1)-lambda/m*Theta2(:,1);%由於不懲罰偏執單元對應的列，所以把他減掉
% Theta1_grad=Theta1_grad/m+lambda/m*Theta1;
% Theta1_grad(:,1)=Theta1_grad(:,1)-lambda/m*Theta1(:,1);

X = [ones(m, 1) X];
ylabel = zeros(num_labels, m);
for i=1:m
    ylabel(y(i), i) = 1;
end

z2 = X*Theta1';
z2 = [ones(m, 1) z2];
a2 = sigmoid(X*Theta1');
a2 = [ones(m, 1) a2];
a3 = sigmoid(a2*Theta2');

for i=1:m
    J = J - log(a3(i, :))*ylabel(:, i) - (log(1 - a3(i, :)) * (1 - ylabel(:, i)));
end
J = J/m;

J = J + lambda/(2*m) * (sum(sum(Theta1(:, 2:end).^2)) + sum(sum(Theta2(:, 2:end).^2)));

Delta1 = zeros(size(Theta1));
Delta2 = zeros(size(Theta2));
for t = 1:m
    delta3 = a3(t, :)' - ylabel(:, t);
    delta2 = Theta2'*delta3 .* sigmoidGradient(z2(t, :)');

    Delta1 = Delta1 + delta2(2:end) * X(t, :);
    Delta2 = Delta2 + delta3 * a2(t, :);
end

Theta1_grad = Delta1 / m;
Theta1_grad(:, 2:end) = Theta1_grad(:, 2:end) + lambda/m*Theta1(:, 2:end);
Theta2_grad = Delta2 / m;
Theta2_grad(:, 2:end) = Theta2_grad(:, 2:end) + lambda/m*Theta2(:, 2:end);
% -------------------------------------------------------------

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];


end

隨機初始化
randInitializeWeights.m

function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
%   W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights 
%   of a layer with L_in incoming connections and L_out outgoing 
%   connections. 
%
%   Note that W should be set to a matrix of size(L_out, 1 + L_in) as
%   the first column of W handles the "bias" terms
%

% You need to return the following variables correctly 
W = zeros(L_out, 1 + L_in);

% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
%               training the neural network.
%
% Note: The first column of W corresponds to the parameters for the bias unit
%
epsilon = 0.12;
W = rand(L_out, 1+L_in)*2*epsilon - epsilon;
% =========================================================================

end

sigmoid梯度
sigmoidGradient.m

function g = sigmoidGradient(z)
%SIGMOIDGRADIENT returns the gradient of the sigmoid function
%evaluated at z
%   g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function
%   evaluated at z. This should work regardless if z is a matrix or a
%   vector. In particular, if z is a vector or matrix, you should return
%   the gradient for each element.

g = zeros(size(z));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
%               each value of z (z can be a matrix, vector or scalar).

g=sigmoid(z).*(1-sigmoid(z));
% =============================================================
end